I think most of us would agree it's a typical G.I. fable, but at its root based on some kind of mis-assumption by a soldier.

One possibility:
The 'story' accompanying the photo says that the saddle was found in a stable of horses near 'Hitler's mansion'.

The only 'Hitler's mansion' most of us can think of was at Obersalzberg.

Obersalzberg seems to be on the known route that this soldier's unit took.

There are several references that say that when Obersalzberg was expanded to include administration buildings, a post office, SS barracks, etc., it also included... horse stables (though information on those stables is pretty much impossible to find).

One reference even refers to an SS Horse Farm at Obersalzberg...although I'm skeptical about that.

Apparently Bormann had farms on Obersalzberg where the later U.S. Army Ski area was located which included all kinds of animals including horses... so that's a possibility.

Since the SS unit assigned to the Obersalzberg for guard duty was part of the Leibstandarte Adolf Hitler, it stands to reason that any horse saddles they might have had, could have featured their unit name, just like their cuff sleeves said Adolf Hitler on it.

The last step then, could have been when a lowly American soldier discovered a saddle with 'Adolf Hitler' on it in the vicinity of the Berghof, he could easily have misconstrued that the nearest horse was 'Hitler's Horse'. At least it made for the whimsical photo, just like soldiers liked to pose in front of Nazi monuments holding a comb over their top lip and making a Hitler salute--- they were just joking around.

Of course surviving relatives now tell the story as if every detail is gospel truth, like the part about the other soldiers wanting to shoot all the horses, but their relative intervened and saved them. In another generation or two they'll have Hitler being an accomplished equestrian rider.

A more accurate conclusion might have been that this might have been 'Hitler's guard's horse'- but even that is a bit of a stretch.